Loosely-Stabilizing Leader Election on Arbitrary Graphs in Population Protocols
In the population protocol model Angluin et al. proposed in 2004, there exists no self-stabilizing protocol that solves leader election on complete graphs without knowing the exact number of nodes. To circumvent the impossibility, we previously introduced the concept of loose-stabilization, which relaxes the closure requirement of self-stabilization. A loosely-stabilizing protocol guarantees that starting from any initial configuration a system reaches a loosely-safe configuration, and after that, the system keeps its specification (e.g. the unique leader) not forever, but for a sufficiently long time. Our previous work presented a loosely-stabilizing protocol that solves the leader election on complete graphs using only the upper bound N of n, not the exact value of n. We take this work one step further in this paper: We propose two loosely-stabilizing protocols that solve leader election for arbitrary graphs. One is a deterministic protocol that uses the identifiers of nodes while the other is a probabilistic protocol that works on anonymous networks. Given the upper bounds N and Δ of the number of nodes and the maximum degree of nodes respectively, both protocols keep a unique leader for Ω(Ne N ) expected steps after entering a loosely-safe configuration. The former enters a loosely-safe configuration within O(mΔN logn) expected steps while the latter does within O(mΔ2 N 3logN) expected steps where m is the number of edges of the graph.
KeywordsLoose-stabilization Population protocols Leader election
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